Lipschitz free $p$-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction

TL;DR

This paper introduces the Schur $p$-property for $0<p extless 1$ and demonstrates that Lipschitz free $p$-spaces over discrete metric spaces have the approximation property, advancing understanding in non-locally convex quasi-Banach spaces.

Contribution

It defines the Schur $p$-property for $0<p extless 1$ and proves Lipschitz free $p$-spaces over discrete metric spaces have the approximation property.

Findings

Introduction of the Schur $p$-property and strong Schur $p$-property for $0<p extless 1$

Development of an adapted compact reduction principle

Proof that Lipschitz free $p$-spaces over discrete metric spaces have the approximation property

Abstract

The geometric analysis of non-locally convex quasi-Banach spaces presents rich and nuanced challenges. In this paper, we introduce the Schur $p$-property and the strong Schur $p$-property for $0 < p \leq 1$, providing new tools to deepen the understanding of these spaces, and the Lipschitz free $p$-spaces in particular. Moreover, by developing an adapted version of the compact reduction principle, we prove that Lipschitz free $p$-spaces over discrete metric spaces possess the approximation property, thereby answering positively a question raised by Albiac et al. in arXiv:2005.06555v2.